[Paper Review] Extended deformation functors

Marco Manetti|ArXiv.org|Oct 14, 1999

Homotopy and Cohomology in Algebraic Topology10 references13 citations

TL;DR

This paper introduces extended deformation functors—generalized deformation theories compatible with Derived Deformation Theory—using Schlessinger-type conditions. It establishes an obstruction theory, proves the inverse mapping theorem for natural transformations, and shows that all such functors with finite-dimensional tangent spaces are prorepresentable in the homotopy category, unifying recent advances in deformation theory.

ABSTRACT

We introduce a precise notion, in terms of few Schlessinger's type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category.

Motivation & Objective

• To formalize a notion of extended deformation functors compatible with modern Derived Deformation Theory (DDT).
• To develop an obstruction theory and analogue of Schlessinger's theory for these extended functors.
• To establish conditions under which such functors are prorepresentable in the homotopy category.
• To prove the inverse mapping theorem for natural transformations between extended deformation functors.
• To unify geometric examples and recent theoretical developments in deformation theory under a single coherent framework.

Proposed method

• Introduce extended deformation functors via a set of Schlessinger-type conditions tailored for derived settings.
• Define the tangent space and obstruction space for extended functors, generalizing classical deformation theory.
• Construct a natural transformation between extended deformation functors and prove its inverse mapping theorem.
• Use homotopical algebra techniques to analyze prorepresentability in the homotopy category.
• Verify that the proposed framework is compatible with known geometric examples and recent DDT developments.
• Apply the theory to show that any extended deformation functor with finite-dimensional tangent space is prorepresentable in the homotopy category.

Experimental results

Research questions

• RQ1 How can deformation functors be extended to be compatible with Derived Deformation Theory while preserving key structural properties?
• RQ2 What conditions ensure that a deformation functor with finite-dimensional tangent space is prorepresentable in the homotopy category?
• RQ3 Can an obstruction theory be developed for extended deformation functors analogous to Schlessinger’s classical theory?
• RQ4 Does the inverse mapping theorem hold for natural transformations between extended deformation functors?
• RQ5 How do the proposed extended functors relate to and generalize existing geometric and algebraic examples?

Key findings

• Extended deformation functors satisfying the proposed Schlessinger-type conditions admit a well-defined obstruction theory.
• The inverse mapping theorem holds for natural transformations between extended deformation functors.
• All extended deformation functors with finite-dimensional tangent space are prorepresentable in the homotopy category.
• The framework is compatible with the main ideas and constructions of Derived Deformation Theory.
• The theory unifies and generalizes classical deformation theory and recent developments in the derived setting.
• The results are formally established via homotopical and categorical techniques, with a journal publication confirming validity.